Solve y=-x^2-22x-96

Question

Function

Find the vertex

Find the axis of symmetry

Rewrite in vertex form

(- 11, 25)

Evaluate

y = - x^{2} - 22 x - 96

Find the x -coordinate of the vertex by substituting a= - 1 and b= - 22 into x = - \frac{b}{2 a}

x = - \frac{- 22}{2 ( - 1 )}

Solve the equation for x

x = - 11

Find the y-coordinate of the vertex by evaluating the function for x = - 11

y = - (- 11)^{2} - 22 (- 11) - 96

Calculate

More Steps

Evaluate

- (- 11)^{2} - 22 (- 11) - 96

Multiply the numbers

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Evaluate

- 22 (- 11)

Multiplying or dividing an even number of negative terms equals a positive

22 \times 11

Multiply the numbers

242

- (- 11)^{2} + 242 - 96

Evaluate the power

- 121 + 242 - 96

Calculate the sum or difference

25

y = 25

Solution

(- 11, 25)

Show Solution

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

Evaluate

y = - x^{2} - 22 x - 96

To test if the graph of y = - x^{2} - 22 x - 96 is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - (- x)^{2} - 22 (- x) - 96

Simplify

More Steps

Evaluate

- (- x)^{2} - 22 (- x) - 96

Multiply the numbers

- (- x)^{2} + 22 x - 96

Rewrite the expression

- x^{2} + 22 x - 96

- y = - x^{2} + 22 x - 96

Change the signs both sides

y = x^{2} - 22 x + 96

Solution

Not symmetry with respect to the origin

Show Solution

Identify the conic

Find the standard equation of the parabola

Find the vertex of the parabola

Find the focus of the parabola

(x + 11)^{2} = - (y - 25)

Evaluate

y = - x^{2} - 22 x - 96

Swap the sides of the equation

- x^{2} - 22 x - 96 = y

Move the constant to the right-hand side and change its sign

- x^{2} - 22 x = y - (- 96)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

- x^{2} - 22 x = y + 96

Multiply both sides of the equation by - 1

(- x^{2} - 22 x) (- 1) = (y + 96) (- 1)

Multiply the terms

More Steps

Evaluate

(- x^{2} - 22 x) (- 1)

Use the the distributive property to expand the expression

- x^{2} (- 1) - 22 x (- 1)

Multiplying or dividing an even number of negative terms equals a positive

x^{2} - 22 x (- 1)

Multiply the numbers

x^{2} + 22 x

x^{2} + 22 x = (y + 96) (- 1)

Multiply the terms

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Evaluate

(y + 96) (- 1)

Apply the distributive property

y (- 1) + 96 (- 1)

Multiplying or dividing an odd number of negative terms equals a negative

- y + 96 (- 1)

Simplify

- y - 96

x^{2} + 22 x = - y - 96

To complete the square, the same value needs to be added to both sides

x^{2} + 22 x + 121 = - y - 96 + 121

Use a^{2} + 2 ab + b^{2} = (a + b)^{2} to factor the expression

(x + 11)^{2} = - y - 96 + 121

Add the numbers

(x + 11)^{2} = - y + 25

Solution

(x + 11)^{2} = - (y - 25)

Show Solution

Solve the equation

x = - 11 + 25 - y x = - 11 - 25 - y

Evaluate

y = - x^{2} - 22 x - 96

Swap the sides of the equation

- x^{2} - 22 x - 96 = y

Move the expression to the left side

- x^{2} - 22 x - 96 - y = 0

Multiply both sides

x^{2} + 22 x + 96 + y = 0

Substitute a= 1,b= 22 and c= 96 + y into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{- 22 \pm 2 2 ^{2} - 4 ( 96 + y )}{2}

Simplify the expression

More Steps

Evaluate

2 2^{2} - 4 (96 + y)

Multiply the terms

More Steps

Evaluate

4 (96 + y)

Apply the distributive property

4 \times 96 + 4 y

Multiply the numbers

384 + 4 y

2 2^{2} - (384 + 4 y)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

2 2^{2} - 384 - 4 y

Evaluate the power

484 - 384 - 4 y

Subtract the numbers

100 - 4 y

x = \frac{- 22 \pm 100 - 4 y}{2}

Simplify the radical expression

More Steps

Evaluate

100 - 4 y

Factor the expression

4 (25 - y)

The root of a product is equal to the product of the roots of each factor

4 \times 25 - y

Evaluate the root

More Steps

Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

2 25 - y

x = \frac{- 22 \pm 2 25 - y}{2}

Separate the equation into 2 possible cases

x = \frac{- 22 + 2 25 - y}{2} x = \frac{- 22 - 2 25 - y}{2}

Simplify the expression

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Evaluate

x = \frac{- 22 + 2 25 - y}{2}

Divide the terms

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Evaluate

\frac{- 22 + 2 25 - y}{2}

Rewrite the expression

\frac{2 ( - 11 + 25 - y )}{2}

Reduce the fraction

- 11 + 25 - y

x = - 11 + 25 - y

x = - 11 + 25 - y x = \frac{- 22 - 2 25 - y}{2}

Solution

More Steps

Evaluate

x = \frac{- 22 - 2 25 - y}{2}

Divide the terms

More Steps

Evaluate

\frac{- 22 - 2 25 - y}{2}

Rewrite the expression

\frac{2 ( - 11 - 25 - y )}{2}

Reduce the fraction

- 11 - 25 - y

x = - 11 - 25 - y

x = - 11 + 25 - y x = - 11 - 25 - y

Show Solution

Rewrite the equation

r = \frac{- sin ( θ ) - 22 cos ( θ ) + 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )} r = - \frac{sin ( θ ) + 22 cos ( θ ) + 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Evaluate

y = - x^{2} - 22 x - 96

Move the expression to the left side

y + x^{2} + 22 x = - 96

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

sin (θ) \times r + (cos (θ) \times r)^{2} + 22 cos (θ) \times r = - 96

Factor the expression

cos^{2} (θ) \times r^{2} + (sin (θ) + 22 cos (θ)) r = - 96

Subtract the terms

cos^{2} (θ) \times r^{2} + (sin (θ) + 22 cos (θ)) r - (- 96) = - 96 - (- 96)

Evaluate

cos^{2} (θ) \times r^{2} + (sin (θ) + 22 cos (θ)) r + 96 = 0

Solve using the quadratic formula

r = \frac{- sin ( θ ) - 22 cos ( θ ) \pm ( sin ( θ ) + 22 cos ( θ ) ) ^{2} - 4 cos ^{2} ( θ ) \times 96}{2 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) - 22 cos ( θ ) \pm 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) - 22 cos ( θ ) + 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 22 cos ( θ ) - 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Solution

r = \frac{- sin ( θ ) - 22 cos ( θ ) + 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )} r = - \frac{sin ( θ ) + 22 cos ( θ ) + 1 + 99 cos ^{2} ( θ ) + 22 sin ( 2 θ )}{2 cos ^{2} ( θ )}

Show Solution

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Graph